State space models (SSMs) have emerged as a powerful framework for modelling long-range dependencies in sequence data. Unlike traditional recurrent neural networks (RNNs) and convolutional neural networks (CNNs), SSMs offer a structured and stable approach to sequence modelling, leveraging principles from control theory and dynamical systems. However, a key challenge in sequence modelling is compressing long-term dependencies into a compact hidden state representation without losing critical information. In this paper, we develop a rigorous mathematical framework for understanding memory compression in selective state space models. We introduce a selective gating mechanism that dynamically filters and updates the hidden state based on input relevance, allowing for efficient memory compression. We formalize the trade-off between memory efficiency and information retention using information-theoretic tools, such as mutual information and rate-distortion theory. Our analysis provides theoretical bounds on the amount of information that can be compressed without sacrificing model performance. We also derive theorems that prove the stability and convergence of the hidden state in selective SSMs, ensuring reliable long-term memory retention. Computational complexity analysis reveals that selective SSMs offer significant improvements in memory efficiency and processing speed compared to traditional RNN-based models. Through empirical validation on sequence modelling tasks such as time-series forecasting and natural language processing, we demonstrate that selective SSMs achieve state-of-the-art performance while using less memory and computational resources.

Detecting emotions in limited text datasets from under-resourced languages presents a formidable obstacle, demanding specialized frameworks and computational strategies. This study conducts a thorough examination of deep learning techniques for discerning emotions in short English texts. Deep learning approaches employ transfer learning and word embedding, notably BERT, to attain superior accuracy. To evaluate these methods, we introduce the "SmallEnglishEmotions" dataset, comprising 6372 varied short English texts annotated with five primary emotion categories. Our experiments reveal that transfer learning and BERT-based text embedding outperform alternative methods in accurately categorizing the text in the dataset.